a simple construction of the ree groups of type 2f4

Journal of Algebra 323 (2010) 1468–1481

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Journal of Algebra

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A simple construction of the Ree groups of type 2 F 4

Robert A. Wilson

School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 June 2009Available online 20 November 2009Communicated by Gernot Stroth

Keywords:Simple groupsGroups of Lie typeRee groupsGeneralised octagon

We give an elementary construction of Ree’s family of finite simplegroups 2 F4(q), avoiding the need for the machinery of Lie algebras,algebraic groups, or buildings. We calculate the group orders andprove simplicity from first principles. Moreover, this is a practicalconstruction in the sense that it gives an explicit description ofthe generalized octagon, and from it generators for many of themaximal subgroups may be easily obtained.

© 2009 Elsevier Inc. All rights reserved.

Introduction

The last infinite family of finite simple groups to be discovered was the family of large Ree groups2 F4(22n+1), which were constructed by Rimhak Ree [9] in 1961, by twisting the groups F4(22n+1) ofLie type with an outer automorphism (see also [1]). A construction in terms of buildings was given byTits [10], and another in terms of Moufang polygons by Tits and Weiss in Chapter 32 of [11]. However,none of these constructions is easy, and the groups remain somewhat inaccessible and little studiedas a result. It is the aim of the present paper to provide a new construction and existence proof whichis considerably easier and shorter than any of the previous constructions.

My construction was inspired by [12], in which I gave a new construction of the Ree groups oftype 2G2. However, the extra difficulties encountered in 26 dimensions as opposed to 7 forced me tomake further simplifications, which in turn led to improvements in my treatment of the small Reegroups [13] and of the Suzuki groups [14]. Instead of using Lie theory, I describe a new algebraicstructure whose automorphism group is the Ree group. The generalized octagon has a completelynatural definition within this new structure. I then calculate the point stabiliser, count the points andprove transitivity, in order to compute the group order and prove simplicity (for n > 0).

E-mail address: [email protected].

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R.A. Wilson / Journal of Algebra 323 (2010) 1468–1481 1469

On the way, I construct many of the maximal subgroups, and for the benefit of Lie theorists pointout the relationships to the long and short root elements, parabolic subgroups, maximal torus, Weylgroup, and so on.

After this work was done, I became aware of the work of Kris Coolsaet [2–4] who has indepen-dently, and somewhat earlier, come up with a description of the Ree–Tits octagon which is almostidentical in substance to my description below. His motivation is more geometrical than group-theoretical, in particular to provide a practical framework for working with the geometry, and to solvevan Maldeghem’s Problem 10 for generalized polygons [8]. He does not consider the main problemwhich I solve here, namely to give an elementary existence proof for the Ree groups. However, hiswork already includes explicit formulae for the long and short root elements, and explicit generatorsfor the maximal parabolic subgroups as well as SL2(q) � 2 and Sz(q) � 2.

Even where my work overlaps with Coolsaet’s, I feel that the treatment is sufficiently different tojustify including it here in full.

1. Definitions

Let F = Fq be the field of order q = 22n+1, and let V be the vector space of dimension 27 over Fspanned by vectors wt , w ′

t and w ′′t for −4 � t � 4. We write the typical vector of V as

v =4∑

t=−4

(λt wt + λ′

t w ′t + λ′′

t w ′′t

).

Let W be a vector space of dimension 26, which will be identified either with a subspace of Vby imposing the condition λ0 + λ′

0 + λ′′0 = 0, or with a quotient of V by imposing the condition

w0 + w ′0 + w ′′

0 = 0.Define a quadratic form Q V on V by

Q V (v) = λ0λ′0 + λ0λ

′′0 + λ′

0λ′′0 +

4∑t=1

(λtλ−t + λ′

tλ′−t + λ′′

t λ′′−t

).

Let Q = Q W be the restriction of Q V to W regarded as a subspace of V . The associated symmet-ric bilinear form is B(v, w) = Q (v + w) + Q (v) + Q (w) in the usual way. It is easy to see thatthis quadratic form is non-degenerate, of minus type. (The radical of B V defined by B V (v, w) =Q V (v + w) + Q V (v) + Q V (w) is the 1-space spanned by w0 + w ′

0 + w ′′0.)

Define a cubic form CV on V by

C V (v) =∑

(i, j,k)

λiλ′jλ

′′k +

1∑t=−1

λtλ′tλ

′′t +

4∑t=1

(λtλ0λ−t + λ′

tλ′0λ

′−t + λ′′t λ′′

0λ′′−t

)

where the first sum is over triples (i, j,k) which are cyclic permutations of

±(1,−2,2),±(1,−3,3),±(1,4,−4),±(−4,2,3),±(−4,3,2).

Denote by C = CW the restriction of CV to W , regarded as a subspace of V . The associated symmetrictrilinear form T is defined by

T (u, v, w) = C(u) + C(v) + C(w) + C(u + v) + C(u + w) + C(v + w) + C(u + v + w).

There is now a natural bilinear, commutative, product u ◦ v defined on W by the identity

T (u, v, w) = B(u ◦ v, w)

1470 R.A. Wilson / Journal of Algebra 323 (2010) 1468–1481

for all u, v, w ∈ W . Notice that T (v, v, w) = 0, and therefore v ◦ v = 0 for all v ∈ W . The ‘interesting’part of the multiplication table is given by

◦ w ′′−4 w ′′−3 w ′′−2 w ′′−1 w ′′1 w ′′

2 w ′′3 w ′′

4

w ′−4 w−4 w−3 w−2 w1

w ′−3 w−4 w−3 w−1 w2

w ′−2 w−4 w−2 w−1 w3

w ′−1 w−4 w1 w2 w3

w ′1 w−3 w−2 w−1 w4

w ′2 w−3 w1 w2 w4

w ′3 w−2 w1 w3 w4

w ′4 w−1 w2 w3 w4

and images under the ‘triality map’ wt �→ w ′t �→ w ′′

t �→ wt . The rest of the multiplication is given byw0 ◦ w ′

0 = w ′′0, w ′

0 ◦ wt = w ′′0 ◦ wt = wt and wt ◦ w−t = w0 for t �= 0, and images under triality. All

other products of basis vectors are 0. (Here our notation for W is as a quotient space of V , so a littlecare is needed when calculating Q and C . In particular, the equation w0 ◦ w ′

t = w ′t looks wrong, but

is in fact correct, for if we want W as a subspace of V we must replace w0 by w ′0 + w ′′

0, and theequation becomes (w ′

0 + w ′′0) ◦ w ′

t = w ′t , which is what one would expect.)

Before we define the rest of the algebraic structure, consider the following coordinate permuta-tions, which act both on V , and on W .

ρ = (w−1 w1)(w−2 w2)(w ′−4 w ′−3

)(w ′−2 w ′−1

)(w ′

1 w ′2

)(w ′

3 w ′4

)(

w ′′−4 w ′′−3

)(w ′′−2 w ′′

1

)(w ′′−1 w ′′

2

)(w ′′

3 w ′′4

),

σ = (w0 w ′′

0

)(w−4 w ′′−4

)(w−1 w ′′

1

)(w1 w ′′−1

)(w4 w ′′

4

)(

w−3 w ′′−2

)(w−2 w ′′−3

)(w3 w ′′

2

)(w2 w ′′

3

)(

w ′−3 w ′−2

)(w ′−1 w ′

1

)(w ′

2 w ′3

).

It is easy to verify that the product of these two involutions has order 8, and therefore they generatea dihedral group D of order 16. This will shortly be revealed as the Weyl group. It is also quiteeasy to check that ρ and σ preserve the quadratic form Q and the cubic form C , in the sense thatQ (wρ) = Q (wσ ) = Q (w) and C(wρ) = C(wσ ) = C(w) for all w ∈ W . They both act by permutingthe terms in the definitions. Indeed, there are just seven orbits, of lengths 1, 4, 4, 4, 8, 8, 16, of D onthe 45 terms in the definition of C .

Finally we shall define a new symmetric (partial) product • on W (regarded as a quotient of V ),such that u • v is defined whenever u ◦ v = 0, and satisfies

u • v = v • u,

u • (v + w) = u • v + u • w,

u • (λv) = λ2n(u • v),

v • v = 0.

In order to make it clear that the product is well defined it is useful to define it for all values of uand v , but it must be borne in mind that only the values u • v for which u ◦ v = 0 will necessarily beinvariant under the group. By the twisted linearity laws above it is sufficient to define • on a basis.The ‘interesting’ part of the multiplication is given by


w3 • w4 = w ′3 • w ′

4 = w ′′3 • w ′′

4 = w4,

w ′1 • w ′

2 = w ′′−1 • w ′′2 = w−3 • w4 = w3,

w1 • w4 = w ′2 • w ′

3 = w ′4

and images under the action of D . The only other non-zero terms may be taken to be

w−2 • w2 = w ′−2 • w ′2 = w ′′−2 • w ′′

2 = w0,

w−3 • w3 = w ′−3 • w ′3 = w ′′−3 • w ′′

3 = w ′′0,

w−4 • w4 = w ′−4 • w ′4 = w ′′−4 • w ′′

4 = w ′0

which are not invariant under the action of D . This product is written out in more detail in Ap-pendix A, to facilitate calculations. Note in particular that all ‘mixed’ products are zero, that iswi • w ′

j = wi • w ′′k = w ′

j • w ′′k = 0.

We see that the terms involving a zero subscript are of the form u • v where u ◦ v �= 0. From them,however, we obtain values such as

(w−1 + w−3) • (w1 + w3) = w−3 • w1 + (w−1 • w1 + w−3 • w3) + w−1 • w3

= w ′′−1 + w ′′0 + w ′′

1

which is properly defined, since (w−1 + w−3) ◦ (w1 + w3) = 0.Let W denote the vector space W endowed with the forms Q and C , and the partial product •. Let

R = R(22n+1) be the automorphism group of W, i.e. the group of linear maps g on W which preserveQ , C and •, in the sense that Q (w g) = Q (w) and C(w g) = C(w) for all w , and v g • w g = (v • w)g

whenever v ◦ w = 0. I claim, and shall prove, that R is the Ree group 2 F4(22n+1).A crucial ingredient of the proof is a grading of the coordinates. In fact, we define three gradings

on the coordinates which are compatible with the various products, in a sense to be explained shortly.The grade of w−t is minus the grade of wt , and the positive grades are as follows:

Vector w ′1 w1 w ′′−1 w2 w ′′

2 w3 w ′2 w ′′

3 w ′3 w ′

4 w ′′4 w4

A-grade 1 2 3 4 5 6 7 8 9 10 11 12B-grade 1 1 2 3 4 5 5 6 7 9 10 11C-grade 1 1 2 2 3 3 4 4 5 6 7 8

(Mnemonic: A stands for absolute, B for bullet (product) and C for cubic (form).) Then for all threegradings, all the terms in the quadratic form have total grade 0. The terms in the cubic form havetotal C-grade 0. And the B-grade of v • w is the sum of the C-grades of v and w .

Now define the (A-, B-, C-)grade of an arbitrary vector to be the largest (A-, B-, C-)grade of itsnon-zero coefficients, and define the (A-, B-, C-)leading term of w to be the term with the highest(A-, B-, C-)grade. In particular the ‘leading term’ may depend on which grade is being used, and maynot be uniquely defined. For most purposes, we regard the coordinates as being in the order of theirA-grade, as follows (with some ambiguity in the coordinates with subscript 0):

w−4, w ′′−4, w ′−4, w ′−3, w ′′−3, w ′−2, w−3, w ′′−2, w−2, w ′′1, w−1, w ′−1,

w0,(

w ′0,

)w ′′

0, w ′1, w1, w ′′−1, w2, w ′′

2, w3, w ′2, w ′′

3, w ′3, w ′

4, w ′′4, w4.


2. Remarks on the definitions

The motivation for most of the above definitions comes from consideration of the exceptional Jor-dan algebra. This consists of 3×3 Hermitian matrices over octonions, which we write in the shorthandnotation

(x, y, z | X, Y , Z) =( x Z Y

Z y XY X z

)

where x, y, z are scalars and X, Y , Z are octonions. The cubic form C is a characteristic 2 version ofthe so-called determinant, which defines the exceptional group E6(q). Fixing a particular element ofdeterminant 1, and calling it the identity element 1, defines the Jordan algebra itself. In characteris-tic 2, we may restrict C to the trace 0 part of the exceptional Jordan algebra, and define a quadraticform Q (v) = C(v) + C(1 + v) on this space. This is the same as the form Q defined above, and ◦ isthe Jordan product (modulo the identity element).

Our basis is chosen so that w0, w ′0 and w ′′

0 correspond to diagonal matrices (1,0,0 | 0,0,0),(0,1,0 | 0,0,0) and (0,0,1 | 0,0,0) modulo the identity matrix (or to (0,1,1 | 0,0,0), (1,0,1 | 0,0,0)

and (1,1,0 | 0,0,0) in the subspace of trace 0 matrices); and wi = (0,0,0 | xi,0,0), w ′i = (0,0,0 |

0, xi,0) and w ′′i = (0,0,0 | 0,0, xi), where {x±i | 1 � i � 4} forms a basis for the split form of the

octonion algebra. The triality map then corresponds to a cyclic permutation of the rows and columnsof the matrices: (a,b, c | A, B, C) �→ (c,a,b | C, A, B). The correspondence between the Jordan multi-plication and the octonion multiplication is given by wi w ′

j = w ′′k when xi x j = xk . Moreover, x1 = x−1

and xk = xk for all k �= ±1.Coolsaet’s definitions [2,3] are expressed in terms of a different definition of the determinant, but

amount to the same thing. A similar construction of Q and C (but obviously not •) works in arbitrarycharacteristic, but then one has to introduce some signs into the definitions. This has also been doneby Rylands and Taylor [7].

The 24 coordinates with non-zero subscript can be identified with the short roots of the F4 rootsystem. Here is one possible labelling, where + stands for 1

2 and − stands for − 12 :

Vector Root Vector Root Vector Root

w−4 1000 w ′−4 + + + − w ′′−4 + + + +w−3 0100 w ′−3 + + − + w ′′−3 + + − −w−2 0010 w ′−2 + − + + w ′′−2 + − + −w−1 0001 w ′−1 + − − − w ′′−1 − + + −w1 −(0001) w ′

1 − + + + w ′′1 + − − +

w2 −(0010) w ′2 − + − − w ′′

2 − + − +w3 −(0100) w ′

3 − − + − w ′′3 − − ++

w4 −(1000) w ′4 − − − + w ′′

4 − − −−

The triples which occur in the definition of C and have no 0 subscript are precisely the triples ofshort roots whose sum is 0. In other words, the Jordan product of two short roots is the vectorcorresponding to the sum of these roots whenever this sum is itself a short root.

Now consider the map

⎛⎜⎝

1 1 0 01 −1 0 00 0 1 10 0 1 −1

⎞⎟⎠

which maps short roots to long roots. The • product is defined for any two perpendicular short roots,and its value corresponds to the preimage of their sum under this map.


The • product also takes values w0, w ′0 and w ′′

0 which are less easy to motivate in this context.We justify our definition by the fact that it works. A different justification is given in [15], where weshow that the • product is uniquely determined by the irreducible subgroup SL3(3).

A more formal definition of the • product may be obtained by first noting that the Jordan product◦ is equivalent to a linear map j : W ∧ W → W , where we may write u ◦ v = j(u ∧ v). Similarly,the new product is equivalent to a map π : ker( j) → W which is F2-linear but not F -linear. Indeed,π(λw) = λ2n

w for any w ∈ W ∧ W and any λ ∈ F , and we write u • v = π(u ∧ v). To check that πor • is invariant under a particular linear map, it may be necessary (and sufficient) to check it on abasis for ker( j), which has dimension 299, so this may not be a trivial task!

The grading can be explained by one of the standard orderings on the roots. For example, theC-grade is the inner product of the short root with the vector −(8,3,2,1), and the B-grade is theinner product with −(11,5,3,1). Alternatively, the B-grade of a short root is the C-grade of thecorresponding long root.

3. Some symmetries of the algebra

In this section I shall define some linear maps and prove that they preserve the algebra W. Lateron these maps will be identified with elements of the torus, Weyl group, and root subgroups of theRee group. We have already proved that W is invariant under the Weyl group D of order 16 generatedby the coordinate permutations ρ and σ .

Next define the following diagonal elements, where α and β are arbitrary non-zero elements ofthe field F . The coordinate vectors are eigenvectors, and the eigenvalues on wt are the inverses ofthose on w−t (and similarly for w ′

t and w ′′t ).

Vector Eigenvalue

w−4 α

w ′′−4 β

w ′−4 α2n+1−1β2n+1−1

w ′−3 αβ1−2n+1

w ′′−3 α2n+1β−1

w ′−2 α1−2n+1β

Vector Eigenvalue

w−3 α2n+1−1

w ′′−2 β2n+1−1

w−2 α−1β2n+1

w ′′1 α2−2n+1

β1−2n+1

w−1 α1−2n+1β2−2n+1

w ′−1 αβ−1

Again it is easy to check that these elements preserve Q , C and •. Indeed, they fix each term of Qand C individually. They form what will shortly be revealed as the maximal torus, which is normalisedby the Weyl group already given. Indeed, σ interchanges α and β , so it is easy to check by eye thatit normalises the torus just defined. The other generator ρ fixes α and maps β to β−1α2n+1

so takesonly a little more effort to check.

Next we wish to define a long root element, t say. We regard W as a quotient of V , so thatw0 + w ′

0 + w ′′0 = 0. It is sufficient for the definition of t to say that it fixes w ′

4, w ′−4 and w ′−1 andmaps

w ′1 �→ w ′

1 + w ′0 + w ′−1.

Then the fact that it fixes • implies that it fixes w ′−3 and w ′2, and maps

w ′3 �→ w ′

3 + w ′2,

w ′−2 �→ w ′−2 + w ′−3,

w0 �→ w0 + w ′−1.

It then follows that w ′0 = w ′

2 • w ′−2 + w ′3 • w ′−3 is also fixed. Then the multiplication table in Ap-

pendix A implies that t fixes w−4, w1, w2, w ′′−3, w ′′1, w ′′

4, and maps


w−3 �→ w−3 + w ′′−3,

w−2 �→ w−2 + w ′′−2 + w−3 + w ′′−3,

w−1 �→ w−1 + w ′′1,

w3 �→ w3 + w2,

w4 �→ w4 + w ′′4,

w ′′−4 �→ w ′′−4 + w−4,

w ′′−2 �→ w ′′−2 + w ′′−3,

w ′′−1 �→ w ′′−1 + w1,

w ′′2 �→ w ′′

2 + w2,

w ′′3 �→ w ′′

3 + w3 + w ′′2 + w2.

To check that this element t preserves Q , C and • requires a certain amount of work, but is not asdifficult as it at first appears. To see that t preserves Q , observe that it does the same to the λs as itdoes to the ws, so by substitution we obtain the new value of Q as

Q(

v ′) = (λ0 + λ−1)2 + (

λ′0

)2 + (λ0 + λ−1)λ′0

+ (λ4 + λ′′

4

)λ−4 + λ′

4λ′−4 + λ′′

4

(λ′′−4 + λ−4

)+ (λ3 + λ2)

(λ−3 + λ′′−3

) + (λ′

3 + λ′2

)λ′−3 + λ′′−3

(λ′′

3 + λ3 + λ′′2 + λ2

)+ λ2

(λ−2 + λ′′−2 + λ−3 + λ′′−3

) + λ′2

(λ′−2 + λ′−3

) + (λ′′

2 + λ2)(

λ′′−2 + λ′′−3

)+ λ1

(λ−1 + λ′′

1

) + (λ′

1 + λ′0 + λ′−1

)λ′−1 + λ′′

1

(λ′′−1 + λ1

)= (λ0)

2 + (λ′

0

)2 + λ0λ′0 + λ4λ−4 + λ′

4λ′−4 + λ′′

4λ′′−4 + λ3λ−3 + λ′3λ

′−3 + λ′′3λ′′−3

+ λ2λ−2 + λ′2λ

′−2 + λ′′2λ′′−2 + λ1λ−1 + λ′

1λ′−1 + λ′′

1λ′′−1

= Q (v)

as required. Similarly to check that t fixes C it is sufficient to substitute in the formula, although thecalculations are a little more substantial. Finally we sketch the proof that • is preserved. First look atthe 10-space W ′ = 〈w ′

i, w0〉. Most of the product on this space has been used to obtain the actionof t on the remainder of the space. All that remains is to check the products which lie inside this10-space. There are 16 such products, and they are all very easy to check, for example

(w ′

3 + w ′2

) • (w ′−2 + w ′−3

) = w ′1 + w ′−1 + w ′

0.

Since t fixes W ′ and W ′⊥ , it is immediate that the products w ′i • w j and w ′

i • w ′′k remain zero. It is

also easy to check the six required equations in which the product lies in 〈w0, w ′0〉, such as

w1 • (w−1 + w ′′

1

) + w2 • (w−2 + w−3 + w ′′−2 + w ′′−3

) = w0 + w ′−1.

This leaves the 16-space W ′⊥ , which is a little more difficult. We illustrate the argument by showingthat the products involving w4 are preserved. The following deals with the products with wi :


(w4 + w ′′

4

) • (w4 + w ′′

4

) = 0,(w4 + w ′′

4

) • (w3 + w2) = w4 + w ′′4,(

w4 + w ′′4

) • w2 = w ′′4,(

w4 + w ′′4

) • w1 = w ′4,(

w4 + w ′′4

) • (w0 + w ′−1

) = 0,(w4 + w ′′

4

) • (w−1 + w ′′

1

) = w ′3 + w ′

2,(w4 + w ′′

4

) • (w−2 + w−3 + w ′′−2 + w ′′−3

) = w3 + w2 + w ′′3 + w ′′

2,(w4 + w ′′

4

) • (w−3 + w ′′−3

) = w3 + w2,(w4 + w ′′

4

) • w−4 = w ′0.

Now the products with w ′′i are trickier as the Jordan products are not always zero. Specifically,

w4 ◦ w ′′1 = w ′

4, w4 ◦ w ′′−2 = w ′3, w4 ◦ w ′′−3 = w ′

2 and w4 ◦ w ′′−4 = w ′1. Therefore we need to check

the following sums, rather than individual terms:

w4 • w ′′1 + w3 • w ′′

2,

w4 • w ′′−2 + w3 • w ′′−1,

w4 • w ′′−3 + w2 • w ′′−1,

w4 • w ′′−4 + w−1 • w ′′−1.

Together with the other four products, this gives:

(w4 + w ′′

4

) • w ′′4 = 0,(

w4 + w ′′4

) • (w ′′

3 + w ′′2 + w3 + w2

) = w4 + w ′′4 + w4 + w ′′

4,(w4 + w ′′

4

) • (w ′′

2 + w2) = w ′′

4 + w ′′4,(

w4 + w ′′4

) • w ′′1 + (w3 + w2) • (

w ′′2 + w2

) = w ′2 + w ′

2,(w4 + w ′′

4

) • (w ′′−1 + w1

) = w ′4 + w ′

4,(w4 + w ′′

4

) • (w ′′−2 + w ′′−3

) + (w3 + w2) • (w ′′−1 + w1

) = w2 + w ′′2 + w2 + w ′′

2,(w4 + w ′′

4

) • w ′′−3 + w2 • (w ′′−1 + w1

) = w2 + w2,(w4 + w ′′

4

) • (w ′′−4 + w−4

) + (w−1 + w ′′

1

) • (w ′′−1 + w1

) = w ′0 + w ′

0

all of which evaluate to 0.Similarly we may define the short root element x of order 4 fixing w−4, w3, w−2, w4, and map-

ping

w−1 �→ w−1 + w−2,

w2 �→ w2 + w1 + w0 + w−2.

We then deduce that w−3 = w−4 • w3 is fixed, as is w0 = w3 • w−3 + w4 • w−4, and also


w ′0 �→ w ′

0 + w−1,

w1 �→ w1 + w0 + w−1 + w−2.

Every other basis vector is a product of two of these, so we deduce that

w ′−4 �→ w ′−4 + w ′′−4,

w ′−3 �→ w ′−3 + w ′−4 + w ′′−4,

w ′′−3 �→ w ′′−3 + w ′−3 + w ′′−4,

w ′′−2 �→ w ′′−2 + w ′−2,

w ′′1 �→ w ′′

1 + w ′′−2 + w ′−2,

w ′−1 �→ w ′−1 + w ′′1 + w ′−2,

w ′′−1 �→ w ′′−1 + w ′1,

w ′′2 �→ w ′′

2 + w ′′−1 + w ′1,

w ′2 �→ w ′

2 + w ′′2 + w ′

1,

w ′3 �→ w ′

3 + w ′′3,

w ′4 �→ w ′

4 + w ′3 + w ′′

3,

w ′′4 �→ w ′′

4 + w ′4 + w ′′

3

and w ′−2, w ′1, w ′′−4 and w ′′

3 are fixed. Checking that x preserves • is a little more time-consumingthan checking t , but no more difficult. Notice that in both cases the vectors which are added on tothe listed basis vectors are of strictly smaller grade, with respect to all three gradings.

Coolsaet’s definitions of the long and short root elements rely more on the Lie theory than our def-initions do, but of course are equivalent. Explicit root elements have also been calculated by Howlett,Rylands and Taylor [5], in order to obtain explicit generators for the Ree groups as 26 × 26 matricesin the Magma computer algebra system.

4. The fundamental subgroups SL2(q) and Sz(q)

Consider the group generated by t , σ and the elements of the torus with β = α−1. These elementsact on 〈w ′′

4, w4〉 as the standard generators

(1 01 1

),

(0 11 0

),

(α 00 α−1

)

of SL2(q), and on 〈w−4, w ′′−4〉, 〈w ′′−1, w1〉 and 〈w−1, w ′′1〉 in the same way. On the 2-spaces 〈w ′

2, w ′3〉

and 〈w ′−3, w ′−2〉 they act in the natural action twisted by the field automorphism x �→ x2n+1. On the

4-spaces 〈w ′′−3, w−3, w ′′−2, w−2〉 and 〈w2, w ′′2, w3, w ′′

3〉 the action is the tensor product of these twoactions.

The remaining 6 coordinates are w ′±4, w ′±1, w0 and w ′0. The first two are centralised by the

given elements, and on the 4-space 〈w ′−1, w0, w ′′0, w ′

1〉 we see the tensor product of the natural

representation twisted by the field automorphism x �→ x22n, with itself. In particular, this group is

isomorphic to SL2(q).Notice that conjugating our SL2(q) by σρ gives us another copy of SL2(q) which commutes with

the first. Thus R contains a group SL2(q) � 2 generated by t, σ ,σ ρ and the torus. In fact, if we adjoin


also ρ , then we obtain a maximal subgroup Sp4(q).2. (The complete list of maximal subgroups wasdetermined by Malle [6].)

Similarly, we shall show that the subgroup generated by x, ρ , and the elements of the torus withα = 1, is isomorphic to Sz(q). On the 4-space 〈w ′′

3, w ′3, w ′

4, w ′′4〉 we write down the generators as

4 × 4 matrices as follows:

⎛⎜⎝

1 0 0 01 1 0 01 1 1 01 0 1 1

⎞⎟⎠ ,

⎛⎜⎜⎝

β 0 0 00 β2n+1−1 0 00 0 β1−2n+1

00 0 0 β−1

⎞⎟⎟⎠ ,

⎛⎜⎝

0 0 0 10 0 1 00 1 0 01 0 0 0

⎞⎟⎠ ,

and notice that these are the standard generators for the Suzuki group Sz(q). The given gen-erators act in the same way on the 4-spaces 〈w ′′−4, w ′−4, w ′−3, w ′′−3〉, 〈w ′

1, w ′′−1, w ′′2, w ′

2〉 and〈w ′−2, w ′′−2, w ′′

1, w ′−1〉. Finally, they act in the exterior square of the natural representation on the6-space 〈w−2, w−1, w0, w ′

0, w1, w2〉. In particular this group is Sz(q).Notice that conjugating our Sz(q) by ρσ gives us another copy of Sz(q) which commutes with the

first. Thus R contains a group Sz(q) � 2 generated by x,ρ,ρσ and the torus.

5. The generalized octagon

In order to prove that R is a simple group of order q12(q6 + 1)(q4 − 1)(q3 + 1)(q − 1) we shallconstruct the generalized octagon on which it acts.

Define a point to be a 1-dimensional subspace 〈v〉 of W such that v = v • w for some vector w .Define two points spanned by vectors u and v to be adjacent if u ◦ v = 0 and u • v = 0. A line isa space 〈u, v〉 spanned by two adjacent points 〈u〉, 〈v〉. In this section I shall show that the pointsare just the images under the automorphism group of W of 〈w−4〉, and that the number of them is1 + q + q3 + q4 + q6 + q7 + q9 + q10 = (1 + q)(1 + q3)(1 + q6).

First observe that the leading term of v • w is (up to a scalar multiplication) the product of theleading terms of v and w . Here the leading term is well defined, because if two basis vectors havethe same C-grade, then at most one of them has non-zero •-product with any given basis vector. Inparticular, if v = v • w then the leading term of v is equal to the leading term of v times the leadingterm of w . Thus we read off from the multiplication table that the leading term of v is one of

w±4, w ′′±4, w ′′±3, w±2.

If the leading term is any of w4, w ′′3, w−2 or w ′′−4 we use the group SL2(q) described above,

generated by σ and t , and an element of the torus with αβ = 1, to map v to a vector with a lower-grade leading term. First observe that this group acts on the 2-spaces 〈w4, w ′′

4〉 and 〈w ′′−4, w−4〉 inits natural representation, so acts transitively on the q + 1 subspaces of dimension 1. This deals withthe two cases w4 and w ′′−4.

In the other two cases, the action is on a 4-space, respectively 〈w ′′3, w3, w ′′

2, w2〉 and 〈w−2, w ′′−2,

w−3, w ′′−3〉, as the tensor product of two Frobenius automorphs of the natural representation. Withoutloss of generality, consider the first case. Thus the leading term of v is w ′′

3, and the leading term of wis w ′′−1. Then we can use conjugates of t by elements of the torus to ensure that the coefficient of w ′′

2in v is 0. Since v ◦ w = 0, the coefficients of w2 and w3 in v are also 0. Hence applying σ reducesthe grade of the leading term of v as claimed. [If the coefficient of w ′

2 in v were non-zero, this wouldcreate a term in w ′

3, which would then be the leading term of v . This is a contradiction.]In the remaining cases we use instead the group generated by x and ρ , and the elements of

the torus with α = 1, which, as we have seen, is isomorphic to Sz(q). It acts on the following two4-spaces:

⟨w ′′

4, w ′4, w ′

3, w ′′3

⟩,⟨

w ′′−3, w ′−3, w ′−4, w ′′−4

⟩.


We explain the first case: the other is similar. The leading term of v is w ′′4 and the leading term

of w is w ′′2. First use a conjugate of x by an element of the torus to ensure the coefficient of w ′

4is 0. Then use a conjugate of x2 to ensure the coefficient of w ′

3 is 0. Then we have (modulo lowerterms) v = w ′′

4 +αw ′′3 and w = w ′′

2 +λw ′′−1. Thus v • w = w ′′4 +λw ′

4 +αw ′3 plus lower terms, whence

α = λ = 0. Thus applying ρ reduces the grade of the leading term of v .In the final case, we have to consider the 6-space⟨

w2, w1, w0, w ′0, w−1, w−2

⟩.

The leading term of v is w2 and the leading term of w is w1. Use a conjugate of x to ensure thecoefficient of w1 in v is 0, and then a conjugate of x2 to ensure the coefficient of w0 is 0. Noww ′

0 ◦ w1 = w1 so the coefficient of w ′0 in v is 0. At this point we can apply a conjugate of x2 again

to reduce the coefficient of w−2 to 0, and then ρ reduces the leading term of v , as required. [If thecoefficient of w ′′−1 or w ′

1 in v were non-zero, this would create a term in w ′′2 or w ′

2, which would bethe leading term of v . This is a contradiction.]

We have shown that all the points are images of 〈w−4〉. Moreover, the argument in reverse showsus exactly how many images there are with each leading term. Indeed, there is just one point 〈w−4〉with leading term w−4, and there are q points 〈w ′′−4 + λw−4〉 with leading term w ′′−4. Each of thelatter gives rise to q2 points with leading term w ′′−3, since in the 4-space 〈w ′′−3, w−3, w ′−4, w ′′−4〉 thereare exactly q2 + 1 points of the Suzuki ovoid, one being 〈w ′′−4〉 and the other q2 having leading termw ′′−3. Thus we have altogether q3 points with leading term w ′′−3. At the next stage, we have SL2(q)

acting on the 4-space 〈w−2, w ′′−2, w−3, w ′′−3〉, and the point 〈w ′′3〉 has just q + 1 images under this

group, since it is fixed by the Borel subgroup of index q + 1. Continuing in this way we obtain in total

1 + q + q3 + q4 + q6 + q7 + q9 + q10 = (q6 + 1

)(q3 + 1

)(q + 1)

points, since every time we use SL2(q), we introduce a factor of q, and every time we use Sz(q), weintroduce a factor of q2.

6. The stabiliser of a point

In this section I show that the stabiliser of a point has order q12(q2 + 1)(q − 1)2. In fact thissubgroup has shape q.q4.q.q4.(Sz(q) × Cq−1), and in Lie theoretic terms this is of course a maximalparabolic subgroup. First define two points 〈u, v〉 to be opposite if B(u, v) �= 0. In the previous sectionwe showed that for a fixed point 〈u〉 there are exactly q10 points 〈v〉 which are opposite to it, andthat the stabiliser of 〈u〉 is transitive on them. Picking u = w−4 and v = w4, it now suffices to showthat the subgroup fixing the vectors w−4 and w4 is exactly the fundamental Sz(q) exhibited above.(The stabiliser of the corresponding points 〈w−4〉 and 〈w4〉 is Cq−1 × Sz(q).)

Now there are exactly q2 +1 points which are adjacent to 〈w−4〉 and not opposite to 〈w4〉, namelythe q2 +1 points of the Suzuki ovoid in 〈w ′′−4, w ′−4, w ′−3, w ′′−3〉. Since the fundamental Sz(q) permutesthese points faithfully and transitively, it suffices to show that the subgroup of R which fixes all thesepoints, as well as the vectors w−4 and w4, is trivial.

It is easy to see that any linear map on this 4-space which fixes all the points is a scalar, andthe fact that w ′′−3 • w ′′−4 = w−4 implies that this scalar is the identity. Thus we may assume thatw4, w−4, w ′′−4, w ′−4, w ′−3, w ′′−3 are all centralised. Now it easy to prove that all basis vectors arefixed. For example w0 = w4 ◦ w−4 is fixed, and w ′

2 = w4 ◦ w ′′−3 is fixed, and similarly w ′′2 = w4 ◦ w ′−3

and w ′′−1 = w4 ◦ w ′−4 and w ′1 = w4 ◦ w ′′−4 are fixed. Then w ′′

1 = w ′−4 • w ′2 and w ′−2 = w ′−4 • w ′

1 andw ′−1 = w ′−3 • w ′

2 and w3 = w ′2 • w ′

1 and w ′′−2 = w ′′−4 • w ′′2 are all fixed. We leave the remaining few

coordinates as an exercise for the reader.

7. Properties of R

Finally we are in a position to compute the order of R = R(q) and prove that, provided q > 2, it isa simple group.


Theorem 1. The order of R(q) is q12(q6 + 1)(q4 − 1)(q3 + 1)(q − 1).

Proof. The number of points is (q6 + 1)(q3 + 1)(q + 1), and the order of the stabiliser of a point isq12(q2 + 1)(q − 1)2. �Theorem 2. If q = 22n+1 > 2 then R(q) is simple.

Proof. We saw that the Borel subgroup has orbits of sizes 1,q,q3,q4,q6,q7,q9,q10 on the points, andthese are fused by the point stabiliser into five orbits of sizes 1, q + q3, q4 + q6, q7 + q9 and q10. Theelement σ of the Weyl group interchanges w−4 with w ′′−4 and fuses each orbit with the next, so inany system of imprimitivity, the orbit of size q + q3 cannot be in the same block as the fixed point.Similarly, the element ρσ interchanges w−4 with w−2, and fuses the first, third and fifth orbits, sothe orbit of size q4 + q6 cannot lie in the same block as the fixed point. Finally, σρσρ interchangesw−4 with w ′′

4 and fuses all the five orbits, so the orbit of size q7 + q9 cannot lie in the same block asthe fixed point. Therefore the action of R(q) on the points of the octagon is primitive.

The point stabiliser has a normal abelian subgroup of order q consisting of conjugates of theinvolution x2. These lie inside one of the factors Sz(q) of Sz(q) × Sz(q) inside Sz(q) � 2, and sinceSz(q) is simple (provided q > 2), the group Sz(q) × Sz(q) is generated by conjugates of x2. But thisgroup contains the full stabiliser of two opposite points, so by transitivity on pairs of opposite pointsit follows that the group generated by all conjugates of x2 is the whole of R(q). Clearly x2 is acommutator since it lies in a simple subgroup Sz(q) (here we use once more the fact that q > 2), soR(q) is perfect. Now Iwasawa’s Lemma states that if G is a finite perfect group acting faithfully andprimitively on a set, such that the point stabiliser has a normal abelian subgroup whose conjugatesgenerate G , then G is simple. Since we have verified that R(q) satisfies all these hypotheses (providedq > 2), it follows that R(q) is simple in these cases. �

Finally, I remark that almost everything in this paper, except this last section and anything elsewhich explicitly depends on counting, goes through for infinite fields F of characteristic 2, providedthat F is perfect (which means that the Frobenius endomorphism x �→ x2 is an automorphism) andhas an automorphism τ which squares to the Frobenius automorphism. It is possible to remove therequirement for F to be perfect, at the expense of replacing the multiplication by a co-multiplicationM : W → W ∧ W /im( J ), where J is the Jordan co-multiplication, so that the twisted linearity law inthe finite case becomes M(λv) = λ2n+1

M(v), which generalises to M(λv) = λτ M(v).

Appendix A

The partial product • takes the following values at basis vectors. All products of basis vectorswhich are defined but not listed here, are zero.

• w−4 w−3 w−2 w−1 w1 w2 w3 w4

w−4 w−4 w ′′−4 w ′−4 w ′−3 w ′′−3 w−3

w−3 w−4 w ′−2 w ′′−2 w ′′1 w ′−1 w3

w−2 w ′′−4 w ′−2 w−2 w−1 w ′1 w ′′

3

w−1 w ′−4 w ′′−2 w−2 w1 w ′′−1 w ′3

w1 w ′−3 w ′′1 w−1 w2 w ′′

2 w ′4

w2 w ′′−3 w ′−1 w1 w2 w ′2 w ′′

4

w3 w−3 w ′1 w ′′−1 w ′′

2 w ′2 w4

w4 w3 w ′′3 w ′

3 w ′4 w ′′

4 w4


• w ′−4 w ′−3 w ′−2 w ′−1 w ′1 w ′

2 w ′3 w ′

4

w ′−4 w−4 w ′′−4 w ′−3 w ′−2 w ′′1 w−1

w ′−3 w−4 w ′−4 w ′′−3 w ′′−2 w ′−1 w1

w ′−2 w ′′−4 w ′−4 w−3 w−2 w ′1 w ′′−1

w ′−1 w ′−3 w ′′−3 w−3 w2 w ′′2 w ′

2

w ′1 w ′−2 w ′′−2 w−2 w3 w ′′

3 w ′3

w ′2 w ′′

1 w ′−1 w2 w3 w ′4 w ′′

4

w ′3 w−1 w ′

1 w ′′2 w ′′

3 w ′4 w4

w ′4 w1 w ′′−1 w ′

2 w ′3 w ′′

4 w4

• w ′′−4 w ′′−3 w ′′−2 w ′′−1 w ′′1 w ′′

2 w ′′3 w ′′

4

w ′′−4 w−4 w ′′−4 w ′−2 w ′−4 w ′′−2 w−2

w ′′−3 w−4 w ′−3 w ′′1 w ′′−3 w ′−1 w2

w ′′−2 w ′′−4 w ′−3 w−1 w−3 w ′1 w ′′

2

w ′′−1 w ′−2 w ′′1 w−1 w3 w ′′

3 w ′4

w ′′1 w ′−4 w ′′−3 w−3 w1 w ′′−1 w ′

2

w ′′2 w ′′−2 w ′−1 w3 w1 w ′

3 w ′′4

w ′′3 w−2 w ′

1 w ′′3 w ′′−1 w ′

3 w4

w ′′4 w2 w ′′

2 w ′4 w ′

2 w ′′4 w4

w1 • w−1 + w2 • w−2 = w3 • w−3 + w4 • w−4 = w0,

w ′1 • w ′−1 + w ′

2 • w ′−2 = w ′3 • w ′−3 + w ′

4 • w ′−4 = w0,

w ′′1 • w ′′−1 + w ′′

2 • w ′′−2 = w ′′3 • w ′′−3 + w ′′

4 • w ′′−4 = w0,

w1 • w−1 + w4 • w−4 = w2 • w−2 + w3 • w−3 = w ′0,

w ′1 • w ′−1 + w ′

4 • w ′−4 = w ′2 • w ′−2 + w ′

3 • w ′−3 = w ′0,

w ′′1 • w ′′−1 + w ′′

4 • w ′′−4 = w ′′2 • w ′′−2 + w ′′

3 • w ′′−3 = w ′0,

w1 • w−1 + w3 • w−3 = w2 • w−2 + w4 • w−4 = w ′′0,

w ′1 • w ′−1 + w ′

3 • w ′−3 = w ′2 • w ′−2 + w ′

4 • w ′−4 = w ′′0,

w ′′1 • w ′′−1 + w ′′

3 • w ′′−3 = w ′′2 • w ′′−2 + w ′′

4 • w ′′−4 = w ′′0.

References

[1] R.W. Carter, Simple Groups of Lie Type, Wiley, 1972.[2] K. Coolsaet, Algebraic structure of the perfect Ree–Tits generalized octagons, Innov. Incidence Geom. 1 (2005) 67–131.[3] K. Coolsaet, On a 25-dimensional embedding of the Ree–Tits generalized octagon, Adv. Geom. 7 (2007) 423–452.[4] K. Coolsaet, A 51-dimensional embedding of the Ree–Tits generalized octagon, Des. Codes Cryptogr. 47 (2008) 75–97.[5] R.B. Howlett, L.J. Rylands, D.E. Taylor, Matrix generators for exceptional groups of Lie type, J. Symbolic Comput. 31 (2001)

429–445.[6] G. Malle, Maximal subgroups of 2 F4(q), J. Algebra 139 (1991) 52–69.[7] L.J. Rylands, D.E. Taylor, Constructions for octonion and exceptional Jordan algebras, Des. Codes Cryptogr. 21 (2000) 191–

203.[8] H. van Maldeghem, Generalized Polygons, Monogr. Math., vol. 93, Birkhäuser, Basel, 1998.[9] R. Ree, A family of simple groups associated with the simple Lie algebra of type (F4), Bull. Amer. Math. Soc. 67 (1961)

115–116.[10] J. Tits, Moufang octagons and the Ree groups of type 2 F4, Amer. J. Math. 105 (1983) 539–594.[11] J. Tits, R. Weiss, Moufang Polygons, Springer, 2002.


[12] R.A. Wilson, An elementary construction of the Ree groups of type 2G2, Proc. Edinb. Math. Soc., in press.[13] R.A. Wilson, Another new approach to the small Ree groups, submitted for publication.[14] R.A. Wilson, A new approach to the Suzuki groups, Math. Proc. Cambridge Philos. Soc., in press.[15] R.A. Wilson, A construction of F4(q) and 2 F4(q) in characteristic 2 using SL3(3), in preparation.

a simple construction of the ree groups of type 2f4

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